how compute sum of value for a field using intervals of different field in solr?

You can implement this by using the JSON facet API for range facets. These will allow you to give a sub facet that works with the domain (i.e. the documents in the bucket) from the facet above.

You start by creating a range facet:

RangeFacetMap rangeFacet = new RangeFacetMap(TIMESTAMP, startTime, endTime, gap);

Then you add a sub facet that invokes a sum:

rangeFacet.withStatSubFacet("requests_sum", "sum(requests)")

You send this request to Solr by using:

final JsonQueryRequest request = new JsonQueryRequest()
    .setQuery("*:*")
    .withFacet("timestamps", rangeFacet);
QueryResponse queryResponse = request.process(solrClient, COLLECTION_NAME);

I don't have an instance or documents to test with right now, but this is built from the information under Nester Facet Example in the reference guide.